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Well-posedness of a class of non-homogeneous boundary valueproblems of the Korteweg-de Vries equation on a finite domain∗∗∗

Published online by Cambridge University Press:  10 January 2013

Eugene Kramer ,
Ivonne Rivas  and
Bing-Yu Zhang
Eugene Kramer
Affiliation:
Department of Mathematics, Physics, and Computer Science, Raymond Walters College, University of Cincinnati, Cincinnati, 45236 Ohio, USA. eugene.f.kramer@uc.edu
Ivonne Rivas
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, 45221 Ohio, USA; rivasie@mail.uc.edu; zhangb@ucmail.uc.edu IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brasil.
Bing-Yu Zhang
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, 45221 Ohio, USA; rivasie@mail.uc.edu; zhangb@ucmail.uc.edu
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Abstract

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin andGhidaglia for the Korteweg-de Vries equation posed on a bounded domain(0,L). We show that this class of Initial-Boundary Value Problems islocally well-posed in the classical Sobolev spaceHs(0,L) for s > -3/4, which provides a positive answer to one of the openquestions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001)1463–1492].

Keywords

The Kortweg-de Vries equationwell-posednessnon-homogeneous boundary value problem
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 358 - 384
Copyright
© EDP Sciences, SMAI, 2013

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References

Benjamin, T.B., Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems. Proc. R. Soc. London A 272 (1972) 47–78. Google Scholar
Bona, J.L., Pritchard, W.G. and Scott, L.R., An evaluation of a model equation for water waves. Philos. Trans. Roy. Soc. London Ser. A 302 (1981) 457–510. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane. Trans. Amer. Math. Soc. 354 (2002) 427–490. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., Forced oscillations of a damped korteweg-de Vries equation in a quarter plane. Commun. Partial Differ. Equ. 5 (2003) 369–400. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation on a finite domain. Commun. Partial Differ. Equ. 28 (2003) 1391–1436. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., Conditional and unconditional well posedness of nonlinear evolution equations. Adv. Differ. Equ. 9 (2004) 241–265. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications. Dyn. Partial Differ. Equ. 3 (2006) 1–69. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane. Ann. Henri Poincaré 25 (2008) 1145–1185. Google Scholar
Bona, J.L., Sun, S.M. and Zhang, B.-Y., Nonhomogeneous problem for the Korteweg-de Vries equation in a bounded domain II. J. Differ. Equ. 247 (2009) 2558–2596. Google Scholar
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I : Shrödinger equations. Geom. Funct. Anal. 3 (1993) 107–156. Google Scholar
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II : the KdV-equation. Geom. Funct. Anal. 3 (1993) 209–262. Google Scholar
Bubnov, B.A., Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain. Differ. Equ. 15 (1979) 17–21. Google Scholar
Bubnov, B.A., Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations. Differ. Equ. 16 (1980) 24–30. Google Scholar
Cerpa, E., Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877–899. Google Scholar
Cerpa, E. and Crépeau, E., Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Henri Poincaré 26 (2009) 457–475. Google Scholar
T. Colin and J.-M. Ghidaglia, Un problème aux limites pour l’équation de Korteweg-de Vries sur un intervalle borné (French) [A boundary value problem for the Korteweg-de Vries equation on a bounded interval] Journées équations aux Drives Partielles, Saint-Jean-de-Monts, Exp. No. III, École Polytech., Palaiseau (1997), p. 10.
Colin, T. and Ghidaglia, J.-M., Un problème mixte pour l’équation de Korteweg-de Vries sur un intervalle borné (French) [A mixed initial-boundary value problem for the Korteweg-de Vries equation on a bounded interval]. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 599–603. Google Scholar
Colin, T. and Ghidaglia, J.-M., An initial-boundary-value problem fo the Korteweg-de Vries equation posed on a finite interval. Adv. Differ. Equ. 6 (2001) 1463–1492. Google Scholar
Colin, T. and Gisclon, M., An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation. Nonlinear Anal. 46 (2001) 869–892. Google Scholar
Colliander, J.E. and Kenig, C., The generalized Korteweg-de Vries equation on the half line. Commun. Partial Differ. Equ. 27 (2002) 2187–2266. Google Scholar
Coron, J.-M. and Crépeau, E., Exact boundary controllability of a nonlinear KdV equation with a critical length. J. Eur. Math. Soc. 6 (2004) 367–398. Google Scholar
Faminskii, A.V., On an initial boundary value problem in a bounded domain for the generalized Korteweg-de Vries equation, International Conference on Differential and Functional Differential Equations (Moscow, 1999). Funct. Differ. Equ. 8 (2001) 183–194. Google Scholar
Faminskii, A.V., Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation. Differ. Integral Equ. 20 (2007) 601–642. Google Scholar
Ghidaglia, J.-M., Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time. J. Differ. Equ. 74 (1988) 369–390. Google Scholar
Ghidaglia, J.-M., A note on the strong convergence towards attractors of damped forced KdV equations. J. Differ. Equ. 110 (1994) 356–359. Google Scholar
Holmer, J., The initial-boundary value problem for the Korteweg-de Vries equation. Commun. Partial Differ. Equ. 31 (2006) 1151–1190. Google Scholar
Kappeler, T. and Topalov, P., Global well-posedness of KdV in H -1(T,R). Duke Math. J. 135 (2006) 327–360. Google Scholar
Kato, T., On the Korteweg-de Vries equation. Manuscr. Math. 28 (1979) 89–99. Google Scholar
Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Advances in Mathematics Supplementary Studies, Stud. Appl. Math. 8 (1983) 93–128. Google Scholar
Kenig, C., Ponce, G. and Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. 4 (1991) 323–347. Google Scholar
Kenig, C., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equations via the contraction principle. Commun. Pure Appl. Math. 46 (1993) 527–620 Google Scholar
Kenig, C., Ponce, G. and Vega, L., A bilinear estimate with applicatios to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573–603. Google Scholar
Komornik, V., Russell, D.L. and Zhang, B.-Y., Stabilization de l’equation de Korteweg-de Vries. C. R. Acad. Sci. Paris 312 (1991) 841–843. Google Scholar
Kramer, E.F. and Zhang, B.-Y., Nonhomogeneous boundary value problems for the Korteweg-de Vries equation on a bounded domain. J. Syst. Sci. Complex 23 (2010) 499–526. Google Scholar
Monilet, L., A note on ill-posedness for the KdV equation. Differ. Integral Equ. 24 (2011) 759–765. Google Scholar
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation. Int. Math. Res. Not. (2002) 1979–2005.
L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation : the periodic case. arXiv:10054805V1[Math AP] (2010).
Pazoto, A.F., Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM : COCV 11 (2005) 473–486. Google Scholar
Pazy, A., Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983). Google Scholar
Perla-Menzala, G., Vasconcellos, C.F. and Zuazua, E., Stabilization of the Korteweg-de Vries equation with localized damping. Q. Appl. Math. 60 (2002) 111–129. Google Scholar
Rivas, I., Usman, M. and Zhang, B.-Y., Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-de Vries equation on a finite domain. Math. Control Rel. Fields 1 (2011) 61–81. Google Scholar
Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM : COCV 2 (1997) 33–55. Google Scholar
Rosier, L. and Zhang, B.-Y., Global stabilization of the generalized Korteweg-de Vries equation. SIAM J. Control Optim. 45 (2006) 927–956. Google Scholar
Rosier, L. and Zhang, B.-Y., Control and stabilization of the Korteweg-de Vries equation : recent progresses. J. Syst. Sci. Complex. 22 (2009) 647–682. Google Scholar
Russell, D.L. and Zhang, B.-Y., Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993) 659–676. Google Scholar
Russell, D.L. and Zhang, B.-Y., Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation. J. Math. Anal. Appl. 190 (1995) 449–488. Google Scholar
Tartar, L., Interpolation non linéaire et régularité. J. Funct. Anal. 9 (1972) 469–489. Google Scholar
B.-Y. Zhang, Boundary stabilization of the Korteweg-de Vries equations, Proc. of International Conference on Control and Estimation of Distributed Parameter Systems : Nonlinear Phenomena. Vorau, Styria, Austria (1993). International Series of Numer. Math. 118 (1994) 371–389.
Zhang, B.-Y., A remark on the Cauchy problem for the Korteweg de-Vries equation on a periodic domain. Differ. Integral Equ. 8 (1995) 1191–1204. Google Scholar
Zhang, B.-Y., Analyticity of solutions for the generalized Korteweg de-Vries equation with respect to their initial datum. SIAM J. Math. Anal. 26 (1995) 1488–1513. Google Scholar
Zhang, B.-Y., Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values. J. Func. Anal. 129 (1995) 293–324. Google Scholar
Zhang, B.-Y., Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 37 (1999) 543–565. Google Scholar